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#"2

5

$2 D

#7

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$

T W R2 ! R2

x1 1 C x2 2 A T . / A

A 3 # 5 TT . / D A T R3

A m # n7!A Rm

T T .c1 1Cc2 2/ D c1T . 1/ C c2T . 2/ 1 2

T c1 c2

7!AA

T W Rn ! Rm

Rm

T

T W R2 ! R2

x1

1; : : : ; p Rn T W Rn ! Rn

T . i / D i D 1; : : : ; pT

Rn T . / D¤ Rn

D C tT W Rn ! Rn

R3 P

P D s C t s; t RT W R3 ! R3 P

R3

T . / T . /P

Rn

D .1 " t/ C t

.1 " t/ C t 0 ' t ' 1T

(t = 1) q (1 – t)p + tqx

(t = 0) p

Rn P

a C b 0 ' a ' 1 0 ' b ' 1 T W Rn ! Rm

P TT . / T . /

f W R ! R f .x/ D mx C b

f b D 0

b ¤ 0

f

T W Rn ! Rm

T .x/ D A C A m # n Rm

T ¤

T W Rn ! Rm

f 1; 2; 3g Rn

fT . 1/; T . 2/; T . 3/g

D .x1; x2/ T . / T .x1; x2/

T T .x1; x2/ D.4x1 " 2x2; 3jx2j/

T T .x1; x2/ D.2x1 " 3x2; x1 C 4; 5x2/

T W Rn ! Rm

TT . / D

Rn

T . / T . /c1T . / C c2T . / D c1 c2

T W R3 ! R3

D .x1; x2; x3/ x3 D 0T . / D .x1; x2; "x3/ T

T W R3 ! R3

D .x1; x2; x3/ x2 D 0 T . / D.x1; 0; x3/ T

T T . / D

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664

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7!A

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775 A

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SG

A A A

Au

u

x2

x1

A

The transformation x Ax.

v

v

Ax

xT R2

.4; 1/ .4; "1/ 7!Ax x1

D t t 0 ' t ' 1 T T .t / D t T . /T . /

TT . /

Rn Rm 7!AT A

A Tn # n In

I2 D#

1 00 1

$1 D

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#01

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T R2 R3

T . 1/ D

2

45

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3

5 T . 2/ D

2

4"3

80

3

5

R2

x1

x2

e2 = 01

!"

!"

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D#

x1

x2

$D x1

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$D x1 1 C x2 2

T

T . / D x1T . 1/ C x2T . 2/

D x1

2

45

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3

5 C x2

2

4"3

80

3

5 D

2

45x1 " 3x2

"7x1 C 8x2

2x1 C 0

3

5

T . 1/T . 2/ T . / T . /

A

T . / D%

T . 1/ T . 2/& #

x1

x2

$D A

T W Rn ! Rm

AT . / D A Rn

A m # n j T . j / j

j Rn

A D%

T . 1/ ! ! ! T . n/&

D In D Œ 1 ! ! ! n ! D x1 1 C ! ! ! C xn n

T

T . / D T .x1 1 C ! ! ! C xn n/ D x1T . 1/ C ! ! ! C xnT . n/

D%

T . 1/ ! ! ! T . n/&

2

64x1

:::xn

3

75 D A

A

AT

Rn Rm

A T . / D 3R2

T . 1/ D 3 1 D#

30

$T . 2/ D 3 2 D

#03

$

# #

A D#

3 00 3

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T W R2 ! R2 R2

'

A

#10

$ #''

$ #01

$ #" '

'

$

A D#

' " '' '

$

' D "=2

(– sin !, cos !)

(cos !, sin !)

(1, 0)

(0, 1)

!

! x1

x2

R2

I2 1 2

x2

x1

x2

10

01

Re!ection throughthe x1-axis

01!10

!100!1

0!110

0!1!10

1001

Re!ection throughthe x2-axis

Re!ection throughthe line x2 " x1

x2 " !x1

Re!ection throughthe line x2 " !x1

!10

!10

0!1

0!1

x1

x2

Re!ection throughthe origin

x1

x2

x1

x2

10

0!1

x1

x2

01

!10

x1

x2

10

01

x2 " x1

Horizontalcontractionand expansion

Verticalcontractionand expansion

0k10

01k0

0 ! k ! 1

0 ! k ! 1 k " 1

k " 1

01

10

10

0k

01

0k

k0

k0

x1

x2

x2

x1

x2

x1

x2

x1

Horizontal shear k110

011k

k ! 0 k " 0

01

01

10

k1

k1

1k

1k

Vertical shear

k ! 0 k " 0

x2

k k 10

k

k

x2

x1

x2

x1

x2

x1x1

0100

0010

10

00

01

Projection ontothe x1-axis

Projection ontothe x2-axis

x1

x2

00

x1

x2

T W Rn ! Rm Rm Rm

Rn

T Rm T Rm

T Rn Rm Rm

T . / D T Rn Rm TRm T . / D

m

T is onto m

n

Domain Range

T is not onto m

T

n

Domain Range

m

T

T Rm

T W Rn ! Rm Rm

Rn

T Rm T . / DT

T Rm

Rn T

m

T is not one-to-one T is one-to-one

TT

mn n

0 000

RangeRangeDomain Domain

SGR2 R2 R2

R2

T

A D

2

41 "4 8 10 2 "1 30 0 0 5

3

5

T R4 R3 T

A AR3

A D T R4

R3 A D

T

T W Rn ! Rm TT . / D

P QP Q Q

P PQ

P Q

T T . / D T T . / DT

Rn

T . / D T . / D T

T . " / D T . / " T . / D " D" ¤ T . / D

T W Rn ! Rm AT

T Rn Rm A Rm

T A

PQ Q R P

R

A Rm

Rm A DT . / D T

Rn Rm

T . / D A DT A D

A

T Rn Rm

Rm A

D .x1; x2/ T . / T .x1; x2/T ..x1; x2//

T .x1; x2/ D .3x1 C x2 5x1 C 7x2 x1 C 3x2/ TT R2 R3

T . /T

A

T . / D

2

43x1 C x2

5x1 C 7x2

x1 C 3x2

3

5 D

2

4

3

5#

x1

x2

$D

2

43 15 71 3

3

5#

x1

x2

$

A

T AA

T T R3

A A 3 # 2 A R3 AA A

R3 R3

x2

The transformation T is notonto .3

e2

e1x1

x2

T

T

x3

x1

Span{a1, a2}

a1

a2

T W R2 ! R2

2 2 " :5 1 1

x2 T

1 2

A 7 # 5 T . / D AR5 R7 T T R7

TT

T W R2 ! R4 T . 1/ D .3; 1; 3; 1/ T . 2/ D ."5; 2; 0; 0/

1 D .1; 0/ 2 D .0; 1/

T W R3 ! R2 T . 1/ D .1; 3/ T . 2/ D .4; "7/T . 3/ D ."5; 4/ 1 2 3

3 # 3

T W R2 ! R2 3"=2

T W R2 ! R2 ""=4T . 1/ D .1=

p2; "1=

p2/

T W R2 ! R21

1 " 2 2 2

T W R2 ! R2

1 2 2 C 3 1

T W R2 ! R2 "3"=4

x1 T . 1/ D ."1=p

2; 1=p

2/

T W R2 ! R2 x1

x2 D x1

T W R2 ! R2

2 2 " 2 1 1

x2 D "x1

T W R2 ! R2 x2

"=2

T W R2 ! R2

x1 x2

T

T W R2 ! R2 T . 1/T . 2/

T .2; 1/

T(e1) T(e2)

x1

x2

T W R2 ! R2

A D Œ 1 2! 1 2 #"1

3

$

T

x1

x2

a2

a1

2

4‹ ‹ ‹‹ ‹ ‹‹ ‹ ‹

3

5

2

4x1

x2

x3

3

5 D

2

43x1 " 2x3

4x1

x1 " x2 C x3

3

5

2

4‹ ‹‹ ‹‹ ‹

3

5#

x1

x2

$D

2

4x1 " x2

"2x1 C x2

x1

3

5

Tx1; x2; : : :

T .x1; x2; x3; x4/ D .0; x1 C x2; x2 C x3; x3 C x4/

T .x1; x2/ D .2x2 " 3x1; x1 " 4x2; 0; x2/

T .x1; x2; x3/ D .x1 " 5x2 C 4x3; x2 " 6x3/

T .x1; x2; x3; x4/ D 2x1 C 3x3 " 4x4 .T W R4 ! R/

T W R2 ! R2

T .x1; x2/ D .x1 C x2; 4x1 C 5x2/ T . / D.3; 8/

T W R2 ! R3

T .x1; x2/ D .x1 " 2x2; "x1 C 3x2; 3x1 " 2x2/T . / D ."1; 4; 9/

T W Rn ! Rm

n # n

T W R2 ! R2

' T

T W Rn ! Rm Rm

Rn Rm

A 3 # 2 7!A

Rn Rm

Rn Rm

n # n

R2

R2#

a 00 d

$

a d ˙1

T W Rn ! Rm

Rn Rm

A 3 # 2 7! AR2 R3

T

T W R3 ! R4

T W R4 ! R3

T W Rn ! Rm A

T A

T W Rn ! Rm A

T Rn Rm A

A T W Rn ! Rm

T . / D B

m # n B AT A D B A B

T

T W Rn ! Rm Rn Rm

m n Tm n

S W Rp ! Rn T W Rn ! Rm

7! T .S. //Rp Rm T .S.c C d //

; Rp c d

TT

T R5

R5

2

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2

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4 "7 3 7 56 "8 5 12 "8

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3

77775

1 2 1WEB" 1 T . 1/ D " 1 2 2 " :5 1

x2 1 " 1

10

10

10

!1

Shear transformation Re!ection through the x2-axis

x2

01

!.51

.51

x1

x2 x2

x1 x1

2 2 " :5 1 2 C :5 1 T . 2/ D 2 C :5 1

T

%T . 1/ T . 2/

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T A A

T AA R7

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: : : :

x1 x2 x3

'x1

(!' (

x1

x1

'x1

(!' (

D x1 1

1 2 3

x2 2 x3 3

x2 x3

x1 1 C x2 2 C x3 3 D

2

436 51 13 3352 34 74 450 7 1 3

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5 % ! ! ! %

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x1 x2 x3

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V D RI

V R #I

I1; I2 I3

RI

1 !1 !

1 !1 !

4 !4 !

DC

BA

1 !

3 !

5 volts

20 volts

30 volts

I2

I1

I3

I1

RI

4I1 C 4I1 C 3I1 D .4 C 4 C 3/I1 D 11I1

AB RI 3I2

RI 11I1 " 3I2 C30

11I1 " 3I2 D 30

"3I1 C 6I2 " I3 D 5

"3I1

6I2

"I3 D "1 ! I3

"I2 C 3I3 D "25

"5

11I1 " 3I2 D 30

"3I1 C 6I2 " I3 D 5

" I2 C 3I3 D "25

I1 D I2 DI3 D "8 I3

I1

2

411"3

0

3

5

"1

C I2

2

4"3

6"1

3

5

"2

C I3

2

40

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3

5

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D

2

4305

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3

5

"

1

I1 I1

1

2 3

R D ; R D Œ 1 2 3 ! D

2

4I1

I2

I3

3

5

RR D

R D

2

43000

3

5; R D

2

4050

3

5; R D

2

400

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3

5

B D

A B

I1 " I2 D 3 " 1 D I1

I2 " I3 D

0 1

2; : : : : k

kA 1 D A 0 2 D A 1

kC1 D A k k D 0; 1; 2; : : :

1 2 0

k

k k

r0 s0 0

0 D#

r0

s0

$

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r1

s1

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#r2

s2

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s3

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1 D M 0

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kC1 D M k k D 0; 1; 2; : : :

f 0; 1; 2; : : :g

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k 7! kC1

AA D

B B

I1

I2

I3

I4

1 !

2 !

4 !

3 !

3 !

4 !

1 !

1 !

20 V

30 V

10 V

20 V

10 V

1 !

5 !

1 !

2 !

4 !

I1

I2

I3

I4

4 !

1 !

2 !

2 !

20 V

40 V

10 V

30 V

3 !

1 !

4 !

2 !

3 !

10 VI4I1

I3I23 !

5 !

1 !

7 !

6 !

2 ! 20 V

40 V

30 V

4 !

4 !

I1 I4

I2 I3

I5

2 !

1 !

2 !

1 !

1 !

3 !

50 V 40 V

20 V30 V

1 ! 3 !

4 !

3 !

2 !

3 !

0

0

4

2

4:97:00:03

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3

5

M 0

k k D 1; : : : ; 20

T1 T2 T3 T4

Tk

.20; 27:5; 30; 22:5/

0º

0º

0º

0º

20º 20º

20º 20º

1 2

4 3

(a)

Plate A

10º

10º

40º

40º

0º 0º

10º 10º

1 2

4 3

(b)

Plate B

A D

2

436 51 1352 34 740 7 1:1

3

5; D

2

4x1

x2

x3

3

5; D

2

433453

3

5

n nn

Œ A !Œ C !

A D C D

A DA D

A m # n A DA Rm

Œ A !

A DA B

A D

A m # n A DRm A m

m # n AA D

Rm

n # n A nA n # n

3 # 3 A BA B

A m # n A DA D

A D

A B m # nA Rm B

S D f 1; 2; 3g R3

S

f ; ; gR2

R5

Rm " f ; gR2

Rn

1 2 3 R52

1 3 1 2

f 1; 2; 3g

A 6 # 5 7! AR5 R6

A m # n m7! A

a bax D b

a b

.x; y; ´/

ax C by C c´ D d

R3 a b c

Three planes intersectingin a line

Three planes intersectingin a point

Three planes with nointersection

Three planes with nointersection

(a) (b)

(c) (c')

h k

x1 C 3x2 D k

4x1 C hx2 D 8

"2x1 C hx2 D 1

6x1 C kx2 D "2

4x1 " 2x2 C 7x3 D "5

8x1 " 3x2 C 10x3 D "3

A

TT

b1 b2 b3

2x1 " 4x2 " 2x3 D b1

"5x1 C x2 C x3 D b2

7x1 " 5x2 " 3x3 D b3

f 1; 2; 3g

A

TT

A

A 2 # 3 R2

A 3 # 3 R3

#56

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f.x; y/ W y D 2xgf.x; y/ W y D x=2g

1; 2 R2

A D Œ 1 2! A D

a2

a1

b

x1

x2

2 # 3 AA D R3

2 # 3 AA D R3

3 # 3 AA

A

2

43

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3

5 D

2

4000

3

5

a

'#1a

$;

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a C 2

$(

a; : : : ; f

2

4a00

3

5

2

4bc0

3

5

2

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5

2

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775

2

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775

2

664

def1

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A

A D

2

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3

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f 1; 2; 3; 4g R5

f 1; 2; 3g 4

f 1; 2; 3gf 1; 2g Rn

f 1; 1 C 2g

1; 2; 3 R3

f 1; 2; 3g

T W Rn ! Rm

T . / D T ." / D "T W R3 ! R3

x2 D 0T .x1; x2; x3/ D .x1; "x2; x3/ T

A 3 # 37! A R3 R3

Rn Rn

R2

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a b

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5

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5 D

2

42p

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5 ; a2 C b2 D 1

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